Description

Statistical mechanics is an area of mathematical physics that bridges the gap between microscopic and macroscopic descriptions of large systems of particles. One of the fundamental models from statistical mechanics is the Ising model, a simplified model of magnetization which was discovered by Willhelm Lenz in 1920's. This model consists of discrete variables that represent magnetic dipole moments of atomic spins that can be in one of two states (+1 or -1). These spins are then arranged in some kind of lattice, such as the square grid, and each spin is allowed to interact with its neighboring spin. There is also an external magnetic field and a temperature parameter which governs the spin interactions.

Lenz originally gave the one-dimension Ising model to his PhD student, Ising, to show an abrupt change when increasing the temperature. This is called a phase transition (similar to the behaviour when water turns to steam). Unfortunately, Ising showed the opposite, and his disappointment was so great, that Ising stopped doing mathematics research and became a high school teacher. Later, Onsager showed that the two dimension version of the Ising model does indeed have a phase transition.

The Ising model has huge number of applications in many different fields in physics and mathematics. Indeed, it has application in image restoration, neuroscience as well as being an interesting discrete probability model in its own right.

This project will investigate the Ising model and other closely related statistical mechanical models, mainly from a theoretical perspective, following the first reference below. For those students who enjoy coding, there are opportunities to simulate these statistical mechanical models and draw pictures. Some possible directions could be

• Peierls argument for the Ising Model and Phase transitions;
• Applications of the Ising Model to other fields;
• Simulations of the Ising model and other statistical mechanical models;
• The algebraic or combinatorial approaches to the Ising Model
• Cluster Expansions

Prerequisites

2H Probability is essential

Students taking the 3H Stochastic Processes or 3H Statistical Mechanics courses will find mutual benefit

Resources

For some background on what may be involved, you should:

• look at some of the recommended literature
• do a web search for "Ising Model".

Reading list. The main reference text will be

• S. Friedli and Y. Velenik Statistical Mechanics of Lattice Systems, A Concrete Mathematical Introduction (2017), Cambridge University Press

Get in touch if you would be interested in doing some simulations and/or have any questions!